A couple of thoughts:
Software EQ is the cheapest/easiest solution. Download the driver directly from your sound chipset manufacturer if it's available, ie Realtek has full equalization built in. Otherwise, software EQ will work.
Upgrading your sound card is another option, as you stated. My wife and I each use Sound Blaster Z cards, which we got refurbished on ebay for about $60 each. Great cards, been using them for a few years now. Even has a very powerful headphone amp built in with a dedicated output.
Last is better speakers. Of course, equalization is good regardless of speakers, but equalizing bad speakers is like polishing a turd.
I've had my Klipsch Promedia 2.1s for over 10 years, but I've had to replace the Control Pod, Amp, and Sub speaker in that timeframe. Completely worth it, as they sound quite good for what they are. There are other good computer speakers out there, or you can buy an amp and power any speakers you want with them.
Tons of options!
Current Primary PC:
AMD 5800X | MSI X570 Carbon | G.Skill 32GB DDR4-3600 16-19-19-39 | MSI GamerZ RTX 3070 | ADATA 8200 Pro 1TB NVME | Micron 1100 2TB SSD | Crucial MX500 2TB | Seasonic Gold 750 Watt | Fractal Define S Case w/Tempered Glass | Noctua NH-D15 CPU Cooler | Sound Blaster Z | Noctua 4x 140mm, 1x 120mm Chromax | Pixio PX7 & PX275H Monitors